Neutral-Type Time-Delay Systems: Theoretical Background

Abstract

Time-delay systems belong to the class of Functional Differential Equations (FDE) which are infinite-dimensional, as opposed to the standard Ordinary Differential Equations (ODE). A neutral-type time-delay equation also known as Neutral Functional Differential Equation (NFDE) describes a system in which the rate of change of the state depends not only on the present and past values of the system state but also on the earlier value of the state rate change. The wave equation subject to nonlinear boundary conditions describing the propagating waves traveling along the rod during the drilling process can be transformed through the D’Alembert method into a more handleable representation: a NFDE. This representation allows taking advantage of the wide range of existing tools for the stability analysis of time-delay systems. This chapter provides the main theoretical elements to address the analysis of the qualitative dynamic response and stability of the drilling system in the framework of NFDE.